Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $z \neq 0$. $p = \dfrac{12z + 30}{2z} \div \dfrac{16z + 40}{4z} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{12z + 30}{2z} \times \dfrac{4z}{16z + 40} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ (12z + 30) \times 4z } { 2z \times (16z + 40) } $ $ p = \dfrac {4z \times 6(2z + 5)} {2z \times 8(2z + 5)} $ $ p = \dfrac{24z(2z + 5)}{16z(2z + 5)} $ We can cancel the $2z + 5$ so long as $2z + 5 \neq 0$ Therefore $z \neq -\dfrac{5}{2}$ $p = \dfrac{24z \cancel{(2z + 5})}{16z \cancel{(2z + 5)}} = \dfrac{24z}{16z} = \dfrac{3}{2} $